CHAPTER 1 INTRODUCTION
3.4 Mathematical Formulation and Solution
3.5.1 Comparison with a few available steady state solutions
We will now perform a few checks to ascertain the validity of our developed models. Figs.
3.4, 3.5 and 3.6 show variation of top and side discharges, expressed as ratios with respect to the hydraulic conductivity of soil and thickness of the soil layer, with time for a few flow ponded drainage situations. The plots are for a single ditch drain receiving water from a field having a negligible depth of ponded water over it. The spacing between the adjacent drains is given a very large value, around 100 m (should be theoretically infinite), so that the flow behavior around each ditch can be taken as independent of the flow behavior around neighboring ditches. In all these cases, the ditches are kept empty every time but the magnitude of h=H1 are made to vary from 0.5 m to 3.0 m. As may be observed, both the top and the side discharge fractions converge to a steady state value of around 0.742 in all the plots, irrespective of the depth of the ditches and the anisotropy and specific storage coefficients of the soil. This common value, as obtained from Fukuda’s (1957) and Youngs’
(1994) (see also Fig. 3.7) steady state solutions of the ponded ditch drainage problem, comes out to be 0.743 and 0.742, respectively – a very good match with the value predicted by our derived solutions thereby showing that the proposed analytical models are correctly developed. Further, from experimental observations, Fukuda (1957) found this ratio to be 0.720, a figure quite close to the analytically predicted value. Thus, this close matching of our results with that of Fukuda’s (1957) experimental value can also be considered as an experimental check on our proposed solutions.
It is to be noted that, for the first case, summing Cp(1) with P varying from 15 to 20 and
) 1 (
Amn with M and N varying from 40 to 50, for the second case, summing Bq(2) and Cp(2) with P and Q varying from 15 to 20 and Amn(2) with M and N varying from 40 to 50, and for
Fig. 3.4. Variation of Qtop
proposed solution of the flow problem of Fig. 3.1 running empty) when the other parameters o (theoretically infinite) δ0 =
1 /
=1
y
x K
K (Kx =1m/day
, m/day 1
(Kx = Ky =0.04
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
50 150 250 350 450 550 650
Qtop(1)/2Kh
Time,
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
50 300 550 800 1050
Qtop(1)/2Kh
Time,
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
50 2500 4950 7400 9850
Qtop(1)/2Kh
Time,
(b)
(c) (a)
top(1) 2Kh and Qshalf Kh ratios with time as obtained from the of the flow problem of Fig. 3.1 for different h=H1
when the other parameters of the flow problem are taken as
=0 and (a) Kx Ky =25/1 (Kx =1 m/day , ,
m/day Ky =1m/day), Ss =0.001 m-1 and ( ),
m/day
04 Ss =0.0001m-1
0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
50 450
Qshalf /Kh
Time,
0.70 0.80 0.90 1.00 1.10 1.20 1.30
50 250
Qshalf /Kh
Time,
650 750 850 950 1050 1150 1250 1350 1450 1550
Time, t(s)
1050 1300 1550 1800 2050 2300 2550
Time,t (s)
0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
500 2500 4500
Qshalf /Kh
Time,
9850 12300 14750 17200 19650 22100 24550
Time,t (s)
ratios with time as obtained from the 1 values (i.e., ditches are f the flow problem are taken as Sa(1) =100 m
), m/day 04 .
=0
Ky (b)
and (c) Kx Ky =25/1
450 850 1250 1650
Time,t (s)
450 650 850 1050
Time, t (s)
6500 8500 10500 12500 14500 16500
Time, t (s)
Fig. 3.5. Variation of Qtop
the proposed solution of the flow problem of Fig. 3.2 are running empty) when
m
) 100
2
( =
Sa (theoretically infinite) m/day
02 .
=0
Ky Ss =0.001 m-1
001 .
=0
Ss and (c)Kx
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800
50 5050 10050 15050 20050 25050
Qtop(2) /2Kh
Time,
0.200 0.300 0.400 0.500 0.600 0.700 0.800
50 550 1050 1550
Qtop(2)/2Kh
Time,
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800
50 550 1050 1550 2050 2550
Qtop(2) /2Kh
Time, (a)
(b)
(c)
top(2) 2Kh and Qr(2) =Ql(2) Kh ratios with time as obtained from of the flow problem of Fig. 3.2 for different h=
are running empty) when the other parameters of the flow problem are taken heoretically infinite), δ0 =0
and (a) Kx Ky =25 ,
m
001 -1 (b) Kx Ky =1/1 (Kx =0.5m/day 1
/
=25
Ky (Kx =0.5 m/day , Ky =0.02 m/day
25050 30050 35050 40050 45050 50050 55050 60050
Time, t(s)
1550 2050 2550 3050 3550
Time, t (s)
0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
1000 6000 11000 16000
Qr(2)=Ql(2) /Kh
Time,
0.70 0.80 0.90 1.00 1.10 1.20 1.30
100 600 1100
Qr(2)=Ql(2) /Kh
Time,
2550 3050 3550 4050 4550 5050 5550 6050
Time,t (s)
0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
100 500 900 1300
Qr(2)=Ql(2) /Kh
Time,
ratios with time as obtained from H1
= values (i.e., ditches s of the flow problem are taken as
1 /
25 (Kx =0.5 m/day , ,
m/day Ky =0.5m/day), ),
m/day Ss =0.0001 m-1
16000 21000 26000 31000 36000 41000
Time, t (s)
1100 1600 2100 2600 3100
Time, t (s)
1700 2100 2500 2900 3300 3700 4100
Time,t (s)
Fig. 3.6. Variation of Qtop(
proposed solution of the flow problem of Fig. 3.3 running empty) when the other
(theoretically infinite), δi ),
m/day Ss =0.001 m-1,(b) m and (c)-1 Kx Ky =25/1
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
50 10050 20050 30050 40050 50050
Qtop(3)/2Kh
Time,
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
50 100050 200050 300050 400050
Qtop(3)/2Kh
Time,
0.30 0.40 0.50 0.60 0.70 0.80
50 6050 12050 18050 24050
Qtop(3)/2Kh
Time, (a)
(b)
(c)
2Kh
) 3
( and Qr(3) =Ql(3) Kh ratios with time as obtained from the of the flow problem of Fig. 3.3 for different h=H
when the other parameters of the flow problem are taken
=0 and (a) Kx Ky =25/1 (Kx =0.0254 )Kx Ky =1/1 (Kx =0.0254 m/day ,Ky = 0
0254 . 0
(Kx = m/day , Ky =0.001016 m/day
0.50 2.00 3.50 5.00 6.50 8.00
1000 51000 101000
Qr(3)=Ql(3) /Kh
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00
100 5100
(Qr=Ql) /Kh
0.70 0.90 1.10 1.30 1.50 1.70 1.90 2.10 2.30
500 10500
Qr(3)=Ql(3) /Kh
50050 60050 70050 80050 90050 100050 110050 120050
Time,t (s)
400050 500050 600050 700050 800050 900050 1000050
Time, t(s)
24050 30050 36050 42050 48050 54050 60050
Time, t(s)
ratios with time as obtained from the H1 values (i.e., ditches are s of the flow problem are taken as Sa(3) =100 m
, m/day
0254 Ky =0.001016
0254 .
0 m/day), Ss =0.001 ),
m/day Ss =0.0001 m-1
101000 151000 201000 251000 301000 351000
Time, t (s)
10100 15100 20100 25100
Time,t (s)
20500 30500 40500 50500
Time, t (s)
the third case, summing Bq(3) and Cp(3) with P and Q varying from 15 to 20, Dr(3) with R varying from 30 to 40 and Amn(3) with M and N varying from 40 to 50, are generally sufficient to achieve a good convergence of our series solutions for most flow situations.
Figs. 3.7(a) and 3.7(b) show variations of Qtop(3)/2Kh and Ql(3) =Qr(3) /Kh with water level fraction for an isolated ditch at different times for a few flow situations of Fig. 3.3. As may be observed, our predicted Qtop(3) /2Kh versus (h−H1)/h and Ql(3) =Qr(3) /Kh versus (h−H1)/h steady state profiles obtained from our analytical model for the flow problem of Fig. 3.3 are found to be matching accurately with the corresponding profiles obtained from the steady state solutions of Youngs (1994), thereby showing, once again, the validity of the proposed analytical model for the variably ponded situation (Case 3). Again the results show the changing water storage with time when the head is reduced on lowering the ditch-water levels. For an incompressible soil, the storage coefficient is zero, and the steady state condition occurs instantaneously. Figs. 3.8(a) and 3.8(b) are similar to that of Fig. 3.7, except that the adjacent ditches are now not separated from each other by an infinite distance but are placed maintaining a ratio of Sa(3)/h=3.0; as may be seen, here also our predicted Qtop(3) /2Kh versus (h−H1)/h and Ql(3) =Qr(3) /Kh versus
h H h )/
( − 1 steady state profiles are found to be corresponding exactly with the identical profiles obtained from the steady state analytical solution of Chahar and Vadodaria (2008b), thereby showing again the accuracy of our developed model for the drainage situation of Fig. 3.3.
In Figs. 3.9 and 3.10, we are taking the same ditch drainage examples, one when the field is subjected to zero depth of ponding and the other, when the surface of the soil is covered with a layer of uniform ponded water of thickness one metre, as had been considered by Kirkham (1965), and are comparing the hydraulic heads and the normalized streamlines as predicted by our analytical solutions corresponding to Figs. 3.2 and 3.3, respectively, with the corresponding values obtained from the series solution of Kirkham (1965). From these figures, it is clear that our analytical models could successfully reproduce the steady state
Fig. 3.7. Comparison of Qtop(3)/2Kh versus (h−H1) h and Qr(3) =Ql(3)/Kh versus h
H
h )
( − 1 profiles as obtained from the proposed solution of the flow problem of Fig. 3.3 at different times with the corresponding steady state profiles obtained from the analytical solution of Youngs’ (1994) when the ponding depth is taken as zero and the other parameters of the flow problem are taken as h=2m, Ss =0.001m-1, K =Kx =Ky =0.5 m/day and
m
) 100
3
( =
Sa (theoretically infinite)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0.0 0.2 0.4 0.6 0.8 1.0
Qr(3)=Ql(3) /Kh
(h-H1)/h
50 s 100 s 150 s 200 s 300 s 550 s
Predictions obtained vide Youngs' (1994) solution
Predictions obtained using the proposed analytical solution for the problem Fig. 3.3 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.0 0.2 0.4 0.6 0.8 1.0
Qtop(3) /2Kh
(h-H1)/h
50 s 100 s 200 s 400 s 650 s 1050 s
Predictions obtained vide Youngs' (1994) solution
Predictions obtained using the proposed analytical solution for the problem Fig. 3.3
(a)
(b)
Fig. 3.8. Comparison of Qtop(3)/2Kh versus (h−H1) h and Qr(3) =Ql(3)/Kh versus h
H
h )
( − 1 profiles as obtained from the proposed solution of the flow problem of Fig. 3.3 at different times with the corresponding steady state profiles obtained from the analytical solution of Chahar and Vadodaria (2008b) when the ponding depth is taken as zero and the other parameters of the flow problem are taken as h=2 m, Sa(3) =6m (Sa(3) h=3),
m-1
001 .
=0
Ss and K =Kx =Ky =0.5m/day
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 0.2 0.4 0.6 0.8 1
Qtop(3) /2Kh
(h-H1)/h
50 s 100 s 200 s 300 s 400 500 s 1050 s
Predictions obtained vide Chahar and Vadodaria’s (2008b) solution
(a)
Predictions obtained using the proposed analytical solution for the problem Fig. 3.3
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 0.2 0.4 0.6 0.8 1
Qr(3)=Ql(3)/Kh
(h-H1)/h
50 s 100 s 150 s 250 s 500 s
(b)
Predictions obtained vide Chahar and Vadodaria’s (2008b) solution
Predictions obtained using the proposed analytical solution for the problem Fig. 3.3
Central Line
m 5
m 3
m 1
0 =0 δ
0 1 . 0
2 . 0
4 . 0
6 . 0
7 . 0
8 . 0
9 .
0 1
5 . 0 3 . 0
m 0 1
m 25 .
−0
m 5 .
−0
m 75 .
−0 m
−1 m 5 .
−1 m
−2
Fig. 3.9. Comparison of lines of equal hydraulic head and normalized streamlines as obtained from the proposed steady state solution of the flow problem of Fig. 3.2 with corresponding values obtained from the analytical solution of Kirkham (1965) for an isotropic soil when the flow parameters of Fig. 3.2 are taken as h=5 m, H1 =3 m, H3 =1 m, εa =0.5 m,
m-1
001 .
=0
Ss and δ0 =0
Impervious Layer
* Depth of ponding and height of the ditch bunds are not in scale; all other dimensions are in scale
Steady state normalized streamlines obtained using the proposed solution for the problem Fig. 3.2 Normalized streamlines obtained vide Kirkham’s (1965) solution
Steady state hydraulic head obtained vide Kirkham’s (1965) solution
Steady state hydraulic head obtained using the proposed solution for the problem Fig. 3.2
×
Fig. 3.10. Comparison of lines of equal hydraulic head and normalised streamlines as obtained from the proposed steady state solution of the flow problem of Fig. 3.3 with corresponding values obtained from the analytical solution of Kirkham (1965) for an isotropic soil when the flow parameters of Fig. 3.3 are taken as h=5 m, H1 =3 m, H3 =1 m,
m, 5 .
=0
εa Ss =0.001 m-1 and δi =1m
0 1 . 0
4 . 0
6 . 0
7 . 0
8 . 0
9 . 0
5 . 0 3
. 0
m 0 1
m 1 5 . 0
m 0
m 5 .
−0 m
−1 m
−2 −0.5m
Impervious Layer
×
* Depth of ponding and height of the ditch bunds are not in scale; all other dimensions are in scale
Steady state normalized streamlines obtained using the proposed solution for the problem Fig. 3.3 Normalized streamlines obtained vide Kirkham’s (1965) solution
Steady state hydraulic head obtained vide Kirkham’s (1965) solution
Steady state hydraulic head obtained using the proposed solution for the problem Fig. 3.3 m
5 m 3
m 1
m
=1
δi Central Line
hydraulic heads and streamlines as obtained from Kirkham’s (1965) steady state analytical solution of the fully penetrating ditch drainage problem, thereby providing us with an check on the truthfulness of our developed solutions for the flow situations of Figs. 3.2 and 3.3, respectively.